Mechanical metamaterials.

Mechanical metamaterials, also known as architected materials, are rationally designed composites, aiming at elastic behaviors and effective mechanical properties beyond ('meta') those of their individual ingredients-qualitatively and/or quantitatively. Due to advances in computational science and manufacturing, this field has progressed considerably throughout the last decade. Here, we review its mathematical basis in the spirit of a tutorial, and summarize the conceptual as well as experimental state-of-the-art. This summary comprises disordered, periodic, quasi-periodic, and graded anisotropic functional architectures, in one, two, and three dimensions, covering length scales ranging from below one micrometer to tens of meters. Examples include extreme ordinary linear elastic behavior from artificial crystals, e.g. auxetics and pentamodes, 'negative' effective properties, behavior beyond classical linear elasticity, e.g. arising from local resonances, chirality, beyond-nearest-neighbor interactions, quasi-crystalline mechanical metamaterials, topological band gaps, cloaking based on coordinate transformations and on scattering cancelation, seismic protection, nonlinear and programmable metamaterials, as well as space-time-periodic architectures.

Active exterior cloaking for the two-dimensional Helmholtz equation with complex wavenumbers and application to thermal cloaking.

We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same approach works for complex wavenumbers which makes it applicable to a variety of media, including media with dispersion, loss and gain. Furthermore, by deriving bounds on Graf's addition formulas with complex arguments, we obtain new estimates that allow to quantify the quality of the cloaking effect. We illustrate our results by applying them to achieve active exterior cloaking for the heat equation. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.

Fresnel drag in space-time-modulated metamaterials.

A moving medium drags light along with it as measured by Fizeau and explained by Einstein's theory of special relativity. Here we show that the same effect can be obtained in a situation where there is no physical motion of the medium. Modulations of both the permittivity and permeability, phased in space and time in the form of traveling waves, are the basis of our model. Space-time metamaterials are represented by effective bianisotropic parameters, which can in turn be mapped to a moving homogeneous medium. Hence these metamaterials mimic a relativistic effect without the need for any actual material motion. We discuss how both the permittivity and permeability need to be modulated to achieve these effects, and we present an equivalent transmission line model.

Hybrid topological guiding mechanisms for photonic crystal fibers.

We create hybrid topological-photonic localisation of light by introducing concepts from the field of topological matter to that of photonic crystal fiber arrays. S-polarized obliquely propagating electromagnetic waves are guided by hexagonal, and square, lattice topological systems along an array of infinitely conducting fibers. The theory utilises perfectly periodic arrays that, in frequency space, have gapped Dirac cones producing band gaps demarcated by pronounced valleys locally imbued with a nonzero local topological quantity. These broken symmetry-induced stop-bands allow for localised guidance of electromagnetic edge-waves along the crystal fiber axis. Finite element simulations, complemented by asymptotic techniques, demonstrate the effectiveness of the proposed designs for localising energy in finite arrays in a robust manner.

Isotropic Chiral Acoustic Phonons in 3D Quasicrystalline Metamaterials.

The elastic properties of three-dimensional (3D) crystalline mechanical metamaterials, unlike those of amorphous structures, are generally strongly anisotropic-even in the long-wavelength limit and for highly symmetric crystals. Aiming at isotropic linear elastic wave propagation, we therefore study 3D periodic approximants of 3D icosahedral quasicrystalline mechanical metamaterials consisting of uniaxial chiral metarods. Considering the increasing order of the approximants, we approach nearly isotropic effective speeds of sound and isotropic acoustical activity. The latter is directly connected to circularly polarized 3D metamaterial chiral acoustic phonons-for all propagation directions in three dimensions.

Topological beam-splitting in photonic crystals.

We create a passive wave splitter, created purely by geometry, to engineer three-way beam splitting in electromagnetism in transverse electric and magnetic polarisation. We do so by considering arrangements of Indium Phosphide dielectric pillars in air, in particular we place several inclusions within a cell that is then extended periodically upon a square lattice. Hexagonal lattice structures are more commonly used in topological valleytronics but, as we discuss, three-way splitting is only possible using a square, or rectangular, lattice. To achieve splitting and transport around a sharp bend we use accidental, and not symmetry-induced, Dirac cones. Within each cell pillars are either arranged around a triangle or square; we demonstrate the mechanism of splitting and why it does not occur for one of the cases. The theory is developed and full scattering simulations demonstrate the effectiveness of the proposed designs.

Frequency domain transformation optics for diffusive photon density waves' cloaking.

We make use of transformation optics technique to realize cloaking operation in the light diffusive regime, for spherical objects. The cloak requires spatially heterogeneous anisotropic diffusivity, as well as spatially varying speed of light and absorption. Analytic calculations of Photon's fluence confirm minor role of absorption in reduction of far-field scattering, and a monopole fluence field converging to a constant in the static regime in the invisibility region. The latter is in contrast to acoustic and electromagnetic cloaks, for which the field vanishes inside the core. These results are finally discussed in the context of mass diffusion, where cloaking can be achieved with a heterogeneous anisotropic diffusivity.

Finite element analysis of electromagnetic waves in two-dimensional transformed bianisotropic media.

We analyze the wave propagation in two-dimensional bianisotropic media with the Finite Element Method (FEM). Starting from the Maxwell-Tellegen's equations in bianisotropic media, we derive some system of coupled Partial Differential Equations (PDEs) for longitudinal electric and magnetic field components. These PDEs are implemented in FEM using a solid mechanics formulation. Perfectly Matched Layers (PMLs) are also discussed to model unbounded bianisotropic media. The PDEs and PMLs are then implemented in a finite element software, and transformation optics is further introduced to design some bianisotropic media with interesting functionalities, such as cloaks, concentrators and rotators. In addition, we propose a design of metamaterial with concentric layers made of homogeneous media with isotropic permittivity, permeability and magnetoelectric parameters that mimic the required effective anisotropic tensors of a bianisotropic cloak in the long wavelength limit (homogenization approach). Our numerical results show that transformation based electromagnetic metamaterials can be extended to bianisotropic media.

Constructing metamaterials from subwavelength pixels with constant indices product.

We investigate a two-dimensional metamaterial template constructed from different pixels through a conservation law of effective indices: If the product of refractive indices along the principal axes is invariant for different anisotropic materials in a two-dimensional space, the product of indices of the effective medium remains constant after mixing these materials. Such effective media of constant indices product can be implemented using metamaterial structures. The orientation of the metamaterial structure in a single pixel controls the direction of the principal axis of the effective medium. Different pixels are assembled into an array to obtain reconfigurable anisotropy of the effective medium. These considerations would be useful for constructing reconfigurable metamaterials and transformation media with area-preserving maps.

Single frequency microwave cloaking and subwavelength imaging with curved wired media.

We consider the cloaking properties of electromagnetic wired media deduced from arbitrary coordinate transformations. We propose an interpretation of invisibility via sub-wavelength imaging features. The quality of cloaking is assessed by the level of deformation of the image of a P-shaped source through the stretched wired media: the lesser the image deformation, the more effective the cloaking. We numerically and experimentally demonstrate a tetrahedral wired cloak with longer edge length about 7cm at a frequency of 1GHz (the cloak is thus subwavelength). The wired cloak has two functionalities: it can serve as a high-resolution imaging system over long distances, and it can also perform space transformations such as, but not limited to, cloaking at a single operation frequency.

Directional cloaking of flexural waves in a plate with a locally resonant metamaterial.

This paper deals with the numerical design of a directional invisibility cloak for backward scattered elastic waves propagating in a thin plate (A0 Lamb waves). The directional cloak is based on a set of resonating beams that are attached perpendicular to the plate and are arranged at a sub-wavelength scale in ten concentric rings. The exotic effective properties of this locally resonant metamaterial ensure coexistence of bandgaps and directional cloaking for certain beam configurations over a large frequency band. The best directional cloaking was obtained when the resonators' length decreases from the central to the outermost ring. In this case, flexural waves experience a vanishing index of refraction when they cross the outer layers, leading to a frequency bandgap that protects the central part of the cloak. Numerical simulation shows that there is no back-scattering in these configurations. These results might have applications in the design of seismic-wave protection devices.

Morphing for faster computations in transformation optics.

We propose to use morphing algorithms to deduce some approximate wave pictures of scattering by cylindrical invisibility cloaks of various shapes deduced from the exact computation (e.g. using a finite element method) of scattering by cloaks of two given shapes, say circular and elliptic ones, thereafter called the source and destination images. The error in L(2) norm between the exact and approximate solutions deduced via morphing from the source and destination images is typically less than 2 percent if control points are judiciously chosen. Our approach works equally well for rotators and concentrators, and also unveils some device which we call rotacon since it both rotates and concentrates electromagnetic fields. However, it breaks down for superscatterers (deduced from non-monotonic transforms): the error in L(2) norm is about 25 percent. We stress that our approach might greatly accelerate numerical studies of 2D and 3D cloaks.

Bicephalous transformed media: concentrator versus rotator and cloak versus superscatterer.

Transformational optics allows for unprecedented control of light with cylindrical cloaks, concentrators, rotators and superscatterers. These are made of different heterogeneous anisotropic media. Can one cloak an s-polarized field and concentrate (or rotate) a p-polarized field with the same metamaterial (or vice versa)? We show the answer is positive provided the geometric transforms underpinning these functionalities take the same values on the outer boundary of what we call a bicephalous metamaterial. In this way, one can also make a metallic cylinder appear invisible for one light polarization, and larger for the other.

Finite frequency external cloaking with complementary bianisotropic media.

We investigate the twofold functionality of a cylindrical shell consisting of a negatively refracting heterogeneous bianisotropic (NRHB) medium deduced from geometric transforms. The numerical simulations indicate that the shell enhances their scattering by a perfect electric conducting (PEC) core, whereas it considerably reduces the scattering of electromagnetic waves by closely located objects when the shell surrounds a bianisotropic core. The former can be attributed to a homeopathic effect, whereby a small PEC object scatters like a large one as confirmed by numerics, while the latter can be attributed to space cancellation of complementary bianisotropic media underpinning anomalous resonances counteracting the field emitted by small objects (external cloaking). Space cancellation is further used to cloak a NRHB finite size object located nearby a slab of NRHB with a hole of same shape and opposite refracting index. Such a finite frequency external cloaking is also achieved with a NRHB cylindrical lens. Finally, we investigate an ostrich effect whereby the scattering of NRHB slabs and cylindrical lenses with simplified parameters hide the presence of small electric antennas in the quasi-static limit.

Negative-index materials: a key to "white" multilayer Fabry-Perot.

The use of negative-index materials is highly efficient for tailoring the spectral dispersion properties of a quarter-wavelength Bragg mirror and for obtaining resonant behavior of a multilayer Fabry-Perot cavity over a very large spectral range. An optimization method is proposed and validated on some first promising devices.

Anisotropic conductivity rotates heat fluxes in transient regimes.

We present a finite element analysis of a diffusion problem involving a coated cylinder enabling the rotation of heat fluxes. The coating consists of a heterogeneous anisotropic conductivity deduced from a geometric transformation in the time dependent heat equation. In contrast to thermal cloak and concentrator, specific heat and density are not affected by the transformation in the rotator. Therein, thermal flux diffuses from region of lower temperature to higher temperature, leading to an apparent negative conductivity analogous to what was observed in transformed thermostatics. When a conducting object lies inside the rotator, it appears as if rotated by certain angle to an external observer, what can be seen as a thermal illusion. A structured rotator is finally proposed inspired by earlier designs of thermostatic and microwave rotators.

Exciting graphene surface plasmon polaritons through light and sound interplay.

We propose a concept that allows for efficient excitation of surface plasmon spolaritons (SPPs) on a thin graphene sheet located on a substrate by an incident electromagnetic field. Elastic vibrations of the sheet, which are generated by a flexural wave, act as a grating that enables the electromagnetic field to couple to propagating graphene SPPs. This scheme permits fast on-off switching of the SPPs and dynamic tuning of their excitation frequency by adjusting the vibration frequency (grating period). Potential applications include single molecule detection and enhanced control of SPP trajectories via surface wave patterning of graphene metasurfaces. Analytical calculations and numerical experiments demonstrate the practical applicability of the proposed concept.

Experiments on transformation thermodynamics: molding the flow of heat.

It was recently shown theoretically that the time-dependent heat conduction equation is form invariant under curvilinear coordinate transformations. Thus, in analogy to transformation optics, fictitious transformed space can be mapped onto (meta)materials with spatially inhomogeneous and anisotropic heat-conductivity tensors in the laboratory space. On this basis, we design, fabricate, and characterize a microstructured thermal cloak that molds the flow of heat around an object in a metal plate. This allows for transient protection of the object from heating while maintaining the same downstream heat flow as without object and cloak.

Transformation thermodynamics: cloaking and concentrating heat flux.

We adapt tools of transformation optics, governed by a (elliptic) wave equation, to thermodynamics, governed by the (parabolic) heat equation. We apply this new concept to an invibility cloak in order to thermally protect a region (a dead core) and to a concentrator to focus heat flux in a small region. We finally propose a multilayered cloak consisting of 20 homogeneous concentric layers with a piecewise constant isotropic diffusivity working over a finite time interval (homogenization approach).

Focussing light through a stack of toroidal channels in PMMA.

We propose a transformational design of an axi-symmetric gradient lens for electromagnetic waves. We show that a metamaterial consisting of toroidal air channels of diameters ranging from 23 nm to 190 nm in a matrix of Polymethylmethacrylate (PMMA) allows for a focussing effect of light over a large bandwidth i.e. [600-1000] nm. We finally propose a simplified design of lens allowing for a two-photon lithography implementation.